Wages of Sine

Hi, folks! I’ve been wrestling with right triangles all week, and I can’t help it, I’m addicted to the relationships between opposites, adjacents and hypotenuses!

It started small. I thought it’d be harmless, maybe divide the opposite by the hypotenuse for a handful of angles. You know, for laughs. Then I thought, hey, why not do the same for adjacents and hypotenuses?

But it wasn’t long before I started to notice correlations between these numbers. As the angles increased the opposites over hypotenuses increased even as the adjacents over hypotenuses decreased. That couldn’t be right, could it? And yet… it had to be!

It was only when I considered the case of angle = 0, and compared it to angle = 90, that I started to break out in a cold equine sweat. Opposite over hypotenuse for angle 0 was equal to 0, while adjacent over hypotenuse for angle 0 was equal to 1. And the reverse was true for angle 90! This was madness! And if I had some constantly incrementing number to feed into this system and mapped the two systems onto two orthogonal axes… something crazier even than unicorn magic could happen!